Linearized Coupled Cluster Correction on the Antisymmetric Product

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Linearized Coupled Cluster Correction on the Antisymmetric Product of 1-reference orbital Geminals Katharina Boguslawski, and Paul Woodson Ayers J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.5b00776 • Publication Date (Web): 14 Oct 2015 Downloaded from http://pubs.acs.org on October 15, 2015

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Journal of Chemical Theory and Computation

Linearized Coupled Cluster Correction on the Antisymmetric Product of 1-reference orbital Geminals Katharina Boguslawski∗,† and Paul W. Ayers‡ Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka ˛ 5, 87-100 Toruń, Poland, and Department of Chemistry and Chemical Biology, McMaster University, Hamilton, 1280 Main Street West, L8S 4M1, Canada E-mail: [email protected]

Abstract We present a Linearized Coupled Cluster (LCC) correction based on an Antisymmetric Product of 1-reference orbital Geminals (AP1roG) reference state. In our LCC ansatz, the cluster operator is restricted to double or to single and double excitations, as in standard single-reference CC theory. The performance of the AP1roG-LCC models is tested for the dissociation of diatomic molecules in their lowest-lying singlet state (C2 , F2 , and BN), the symmetric dissociation of the H50 hydrogen chain, and spectroscopic constants of the uranyl cation (UO2+ 2 ). Our study indicates that an LCC correction based on an AP1roG reference function is more robust and reliable than corrections based on perturbation theory, yielding spectroscopic constants that are in very good agreement with theoretical reference data. To whom correspondence should be addressed Toruń ‡ McMaster University

∗

† UMK

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1 Introduction Unlike quantum-chemistry methods for modeling small and medium-sized molecules, 1,2 routine applications to larger molecular systems are hampered because conventional methods are either too expensive or too approximate to guarantee reliable results. These problems are particularly severe for molecules with strong electron correlation. Examples of strongly-correlated systems are radicals, transition metals, and actinide compounds. Furthermore, well-established approaches for strong correlation scale poorly, often factorially, with system size. These drawbacks motivate the development of new, low-cost, electron correlation methods for strongly-correlated many-body systems. One active field of research for strongly-correlated systems focuses on the development of wavefunction methods that use two-electron functions (geminals) to model the correlated motion of electrons. 3–7 The most popular geminal-based approaches are the Antisymmetric Product of Strongly orthogonal Geminals 3,8–14 (APSG), the Antisymmetrized Geminal Power 15–18 (which is a special case of projected Hartree–Fock–Bogoliubov 19 ), the Antisymmetric Product of Interacting Geminals 4,20–31 (APIG), Generalized Valence Bond 8,32–35 (GVB), and the Antisymmetric Product of 1-reference orbital Geminals (AP1roG). 36–38 Specifically, AP1roG provides a very good approximation to the doubly occupied configuration interaction 39 (DOCI) wavefunction, at meanfield computational cost. The AP1roG geminal creation operator reads

φi† = a†i a†i¯ + ∑ cai a†a a†a¯ ,

(1)

a

where a†p , a†p¯ are the standard fermionic creation operators for spin-up (p) and spin-down electrons ( p), ¯ cai are the geminal coefficients, and the summation runs over all virtual orbitals. Specifically,


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the AP1roG geminal coefficient matrix has the form 

cP+1 1

cP+2 1

1 · · · 0 0  0 1 · · · 0 cP+1 cP+2  2 2 cAP1roG =  . . . ..  .. .. . . . .. .   . 0 .. · · · 1 cP+1 cP+2 P P

···



cK 1

  · · · cK 2 . , .. . ..    · · · cK P

(2)

where K denotes the number of basis functions, P the number of electron pairs, and the left subblock of cAP1roG encodes some reference determinant. The geminal matrix connects each geminal with the underlying one-particle basis functions. We should note that if we impose specific restrictions on the structure of the above matrix, we can deduce different geminal models in the APIG family. 5 The electronic wavefunction is written as a product of geminal creation operators for all electron pairs P acting on the vacuum state, |AP1roGi = ∏Pi φi |i. Unique among geminal methods, the AP1roG wavefunction ansatz can be rewritten in terms of one-particle functions as a fully general pair-Coupled-Cluster-Doubles 40 (pCCD) wavefunction,

|AP1roGi = exp

P

K

∑ ∑ i=1 a=P+1

!

cai a†a a†a¯ ai¯ai |Φ0 i,

(3)

where |Φ0 i is some independent-particle wavefunction (for instance the Hartree–Fock (HF) determinant). The exponential ansatz of AP1roG (cf. eq. (3)) ensures the size-extensivity of the model. However, to recover size-consistency, we have to optimize the one-particle basis functions. This can be done in a fully variational manner, 37,38 analogous to orbital-optimized Coupled Cluster, 41 using non-variational orbital optimization techniques 42,43 or a stochastic random-walk algorithm. 44 A number of numerical studies on systems with strongly correlated electrons showed the superiority of the variational orbital optimization procedure over the latter ones. 43 We should note that due to the four-index transformation of the electron repulsion integrals, the computa tional scaling of the orbital-optimized AP1roG model deteriorates to O K 5 . 37 Although restricted 3 ACS Paragon Plus Environment


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orbital-optimized AP1roG is limited to close-shell systems, it has already proven to be a reliable method for modeling strong electron correlation effects in quasi-degenerate systems, 37,42,43 single and multiple bond-breaking processes, 45,46 and actinide chemistry. 47 As all other geminal models, AP1roG misses a large fraction of weak (dynamical) electron correlation effects. To address this problem and account for weak electron correlation effects in the geminal reference wavefunction, various a posteriori corrections have been proposed. These include models based on single- and multi-reference Perturbation Theory, 31,48–51 Extended Random Phase Approximation, 52–54 (Linearized) Coupled Cluster theory, 35,55 and Density Functional Theory. 54,56,57 In the case of AP1roG, dynamical correlation was included using Perturbation Theory, 49 single-reference Coupled-Cluster theory, 58 and Density Functional Theory. 56,57 Recent studies on diatomic molecules, however, point out numerical instabilities and failures of the proposed Perturbation Theory corrections. 46 This motivates the pursuit of more robust dynamical correlation models for AP1roG. A reliable way to account for dynamical correlation effects a posteriori is to use a multi-reference Linearized Coupled Cluster (LCC) correction. Recently, Zoboki et al. presented an LCC correction based on an APSG reference function and demonstrated the good performance of the APSG-MRLCC approach. 55 Their findings encouraged us to develop an LCC correction based on an AP1roG reference state. This work is organized as follows. In section 2, we will discuss two different LCC corrections for AP1roG. Their performance is compared by studying some well-known problems in quantum chemistry that require a balanced treatment of dynamical and strong electron correlation effects: the dissociation of the singlet ground state of C2 and F2 , the dissociation of the first excited singlet state of BN, the symmetric dissociation of H50 , and the spectroscopic constants of the UO2+ 2 molecule. Computational details are presented in section 3, while numerical results are discussed in section 4. Finally, we conclude in section 5.


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2 LCC Theory with an AP1roG Reference Function In this work, dynamical correlation effects are built in the electronic wavefunction a posteriori using an exponential Coupled Cluster ansatz, |Ψi = exp(Tˆ )|AP1roGi,

(4)

where Tˆ = ∑ν tν τˆν is a general cluster operator. The corresponding time-independent Schrödinger equation reads Hˆ exp(Tˆ )|AP1roGi = E exp(Tˆ )|AP1roGi.

(5)

Multiplying from the left by exp(−Tˆ ) and truncating the Baker–Campbell–Hausdorff expansion after the second term, ˆ Tˆ ], exp(−Tˆ )Hˆ exp(Tˆ ) ≈ Hˆ + [H,

(6)

we arrive at the Linearized Coupled Cluster Schrödinger equation ˆ Tˆ ])|AP1roGi = E|AP1roGi. (Hˆ + [H,

(7)

To obtain the cluster amplitudes tν , we multiply from left by hν | ˆ Tˆ ])|AP1roGi = 0, hν |(Hˆ + [H,

(8)

where we assume that the excitation operator τˆν creates states orthogonal to |AP1roGi, hν |AP1roGi = 0. The projection manifold {ν } will depend on the choice of the cluster operator Tˆ (vide infra). The energy can be calculated by projecting against the reference determinant of |AP1roGi, i.e., multiplying eq. (7) by hΦ0 | and using intermediate normalization, ˆ Tˆ ])|AP1roGi = E. hΦ0 |(Hˆ + [H,


(9)


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The only constraint on the cluster operator we have made so far is that it creates states that are orthogonal to the AP1roG reference function. One suitable choice for the cluster operator is to include substitutions between the occupied and virtual orbitals with respect to |AP1roGi. If only double excitations are included, the cluster operator is specified as 1 occ virt Tˆ2 = ∑ ∑ ′tiabj Eâi Eˆb j , 2 i j ab

(10)

where Eâi = a†a ai + a†a¯ ai¯ is the singlet excitation operator and the cluster amplitudes are symmetric with respect to pair-exchange, i.e., tiabj = t ba ji . The prime in the above summations indicates that pair-excited determinants are excluded in the cluster operator, i.e., tiai¯a¯ = 0 (as those excitations do not fulfill the orthogonality condition, hν |AP1roGi = 0). To arrive at a computationally feasible model, we will further restrict the cluster operator of eq. (10) to include only excitations with respect to the reference determinant, thereby excluding possible redundancies in excitations and amplitudes. The projection manifold then contains all doubly-excited determinants with respect to |Φ0 i. Furthermore, as basis for the bra states of the projection manifold, we will use the convenient choice 41 1 ab 1 ab hab i j | = hi j | + h ji |, 3 6

(11)

ˆ ˆ where hab i j | = hΦ0 |E jb Eia . The bra basis then forms a biorthogonal basis that satisfies the normalization condition cd hab i j |kl i = δia jb,kcld + δ jbia,kcld .

(12)

The doubles amplitudes {tiabj } are obtained by solving a linear set of equations Bµ + ∑ Aµ,ν tν = 0,

(13)

ν

ˆ where the sum runs over all double excitations (without pair excitations) and Bµ = Bia jb = hab i j |H|AP1roGi,


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(D) ˆ ˆ ˆ while Aµ,ν = Aia jb,kcld = 12 hab i j |[H, Eck Edl ]|AP1roGi. The energy correction Ecorr with respect to

the AP1roG reference wavefunction is given as (D)

Ecorr =

∑ tiabj (hi j||abi + hi j|abi),

(14)

ia jb

where we have used the standard notation for the exchange integrals, hi j||abi = hi j|abi − hi j|bai, and physicists’ notation for the two-electron integrals. Similarly, the contribution of single excitations can be accounted for by including Tˆ1 = ∑ tia Eâi

(15)

ia

in the cluster operator. Restricting the single excitations to the AP1roG reference determinant |Φ0 i, the singles projection manifold contains all singly-excited determinants with respect to |Φ0 i. In analogy to the doubles projection manifold, the bra states of the singles projection manifold are chosen to form a biorthogonal basis with the convenient normalization condition 41

hai |bj i = δai,b j ,

(16)

where hai | = 12 hai | = 12 hΦ0 |Eîa . The single and double amplitudes are obtained by solving a coupled set of linear equations equivalent to eq. (13) where µ and ν now run over all single and double excitations. The energy correction with respect to the AP1roG reference value is (S,D)

Ecorr = 2 ∑ Fiatia + ∑ tiabj (hi j||abi + hi j|abi), ia

(17)

ia jb

where Fia are the elements of the Fock matrix, Fia = hia + ∑occ m (ham||imi + ham|imi) and hia are the one-electron integrals. Note that, in contrast to canonical Hartree–Fock orbitals, the Fock matrix is not diagonal when the orbitals are optimized within AP1roG. In the AP1roG-LCC approach, the single excitations thus contribute both directly to the energy correction and indirectly through



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coupling to the doubles equations. We will abbreviate the LCC correction using Tˆ = Tˆ2 as AP1roG-LCCD, while AP1roG-LCCSD indicates that the cluster operator contains single and double excitations, Tˆ = Tˆ1 + Tˆ2 . Finally, we should note that the LCCD and LCCSD corrections as outlined above are similar, but not equivalent to the frozen-pCCD (fpCCD) and fpCCSD approaches, respectively. 58 Specifically, in fpCC, first the equations for the pair amplitudes are solved, which in our case is equivalent to solving the AP1roG amplitude equations. Then, the usual CCD/CCSD equations are solved without allowing the pair amplitudes to change. In AP1roG-LCC, we first solve for the AP1roG amplitudes, which is equivalent to the first step of a fpCC calculation, followed by solving for the remaining cluster amplitudes. In contrast to fpCC, our cluster operator is linearized (cf., eq. (7)) and thus all higher-order terms are eliminated, while the reference wavefunction is |AP1roGi (instead of a single Slater determinant as in fpCC). Choosing |AP1roGi as a reference function results in additional terms in the amplitude equations beyond the standard single-reference LCC approach arising from coupling to pair-excited Slater determinants with respect to |Φ0 i. This coupling to pair-excited Slater determinants leads to additional terms in the LCC amplitude equations that are also included in the fpCC amplitude equations. Therefore, our LCC approach can be considered as a simplification of the fpCC method. (Other modifications of conventional CC-type methods that are suitable for strong correlation have been presented recently. 59,60 ) Specifically, the connection between our LCCD corrections and fpCC can also be understood by rewriting the corresponding Linearized Coupled Cluster Schrödinger equation (7) using explicitly the exponential ansatz for AP1roG, ˆ Tˆnp ]) exp(Tˆp )|Φ0 i = E|Φ0 i, exp(−Tˆp )(Hˆ + [H,

(18)

where Tˆp is the AP1roG (seniority zero) cluster operator containing only pair excitations and Tˆnp is the seniority non-zero cluster operator of eq. (10). Note that we have used the labels p (pair) and np (non-pair) to emphasize the connection to fpCC methods. The seniority non-zero cluster


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amplitudes can be obtained by projection against hν |, ˆ Tˆnp ] + [[H, ˆ Tˆnp ], Tˆp ]|Φ0 i = 0, hν |Hˆ + [H,

(19)

which further illustrates the coupling to pair-excited Slater determinants generated by Tˆp (compared to single-reference Linearized Coupled Cluster). The above equation can be compared to the amplitude equations of (fp-)CC approaches in, for instance, Refs. 41,61 Since the most expensive contributions in the amplitude equations are similar in both AP1roG-LCCD/LCCSD and fpCCD/CCSD, the computational scaling of our LCC correction is as O(o2 v4 ), where o and v are the number of occupied and virtual orbitals, respectively.

3 Computational Details 3.1 AP1roG All geminal calculations have been performed in the H ORTON 2.0.0 software package. 62 All restricted (variationally) orbital-optimized AP1roG calculations were allowed to freely relax without any spatial symmetry constraints, i.e., no point group symmetry was imposed. For all molecules, all orbitals were active. In the following, we will abbreviate (variationally) orbital-optimized AP1roG as AP1roG. We should note that AP1roG is a product of natural geminals. Thus the 1-electron reduced density matrix is diagonal and the (optimized) orbitals are natural orbitals. We will refer to the variationally optimized AP1roG orbitals as natural orbitals.

3.2 PTa and PTb The PTa and PTb calculations were performed using the H ORTON 2.0.0 software package. 62 For all molecules, the optimized AP1roG natural orbitals were chosen as the orbital basis and all electrons and orbitals have been included in the active space.



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3.3 LCCD and LCCSD The Linearized Coupled Cluster models with double and single and double excitations have been implemented in a developer version of H ORTON 2.0.0. 62 The natural orbitals of AP1roG were chosen as the orbital basis for all molecules studied. Furthermore, all electrons and orbitals were correlated.

3.4 Coupled Cluster The Coupled Cluster Doubles (CCD), CC Singles and Doubles (CCSD) as well as CC Singles, Doubles and perturbative Triples (CCSD(T)) calculations have been carried out in the DALTON2013 software package. 63 In each case, all electrons and orbitals were correlated and no spatial symmetry was imposed.

3.5 Relativity and Basis Sets For the C2 and F2 molecules, Dunning’s aug-cc-pVDZ (C, F:(10s5p2d) → [4s3p2d]) and aug-ccpVTZ (C, F: (11s6p3d2f) → [5s4p3d2f]) basis sets were used, while Dunning’s cc-pVDZ (B, N: (9s4p1d) → [3s2p1d]) and cc-pVTZ (B, N: (10s5p2d1f) → [4s,3p,2d,1f]) basis sets were used for BN. The STO-6G basis set was utilized for the H atoms in H50 to allow for a comparison to DMRG reference data. For UO2+ 2 , scalar relativistic effects were incorporated through relativistic effective core potentials (RECP). In all calculations, we have used a small core (SC) RECP (60 electrons in the core) with the following contraction scheme (12s11p10d8f) → [8s7p6d4f]. For the lighter elements (O), the cc-pVDZ basis set of Dunning was employed, (10s5p1d) → [4s3p1d].

3.6 Fitting Procedure The potential energy curves of diatomic molecules were obtained by varying bond lengths in a range of 1.2 − 4.0 Å and 1.1 − 3.2 Å for the F2 and C2 molecules, respectively, and in a range 10 ACS Paragon Plus Environment

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Table 1: Spectroscopic constants for the dissociation of the C2 , F2 , and BN molecule for different quantum chemistry methods and basis sets. The differences are with respect to MRCI-SD reference data 66 for C2 and F2 , and with respect to MRCI-SD+Q reference data 67 for BN. Ee : ground state energy at re . Method

C2

F2

BN

Ee [Eh ]

re [Å]

De [ kcal mol ]

De [ kcal mol ]

ωe [cm−1 ]

Ee [Eh ]

1943(+134) 1901(+93) 1863(+55) 1960(+151) 1855(+46) 1924(+117) 1809

−75.58569 −75.80222 −75.78350 −75.81125 −75.81257 −75.78829 −75.78079

aug-cc-pVTZ 1.227(−0.025) 132.9(−8.2) 1.251(+0.001) 160.4(+19.3) 1.235(−0.017) 127.2(−13.9) 1.240(−0.012) 139.3(−1.8) 1.240(−0.012) 143.0(+1.9) 1.244(−0.008) 148.0(+6.9) 1.252 141.1

1780(−56) 1889(+53) 1938(+102) 1916(+80) 1926(+90) 1886(+50) 1836 703(−189) 847(−45) 891(−1) 872(−20) 883(−9) 1004(+12) 911(+19) 892

re [Å]

AP1roG AP1roG-PTa AP1roG-PTb AP1roG-LCCD AP1roG-LCCSD NEVPT2 MRCI-SD 66

−75.55256 −75.73784 −75.70689 −75.73848 −75.73882 −75.71751 −75.73221

aug-cc-pVDZ 1.240(−0.033) 104.6(−15.7) 1.273(+0.000) 152.7(+22.4) 1.260(−0.013) 116.3(−14.0) 1.274(−0.001) 124.0(−6.3) 1.266(−0.007) 127.6(−2.7) 1.259(−0.014) 135.0(+4.7) 1.273 130.3

AP1roG AP1roG-PTa AP1roG-PTb AP1roG-LCCD AP1roG-LCCSD CCSD CCSD(T) MRCI-SD 66

−198.88134 −199.14227 −199.13392 −199.15802 −199.15885 −199.13428 −199.14777 −199.12245

1.521(+0.068) 1.398(−0.055) 1.473(+0.020) 1.466(+0.013) 1.462(+0.009) 1.426(−0.017) 1.450(−0.003) 1.453

12.8(−15.7) 30.1(+1.6) — 39.5(+11.0) 40.1(+11.6) 57.3(+18.8) — 28.5

886(+85) 636(−165) 832(+31) 780(−21) 793(−8) 917(+116) 841(+40) 801

−198.96762 −199.32912 −199.31858 −199.34112 −199.34235 −199.29382 −199.31364 −199.27607

1.467(+0.047) 1.448(+0.028) 1.417(−0.003) 1.433(+0.013) 1.431(+0.011) 1.396(−0.024) 1.419(−0.001) 1.420

AP1roG AP1roG-PTa AP1roG-PTb AP1roG-LCCD AP1roG-LCCSD CCSD(T) 68 RMR CCSD(T) 68 MRCI-SD+Q 67

−79.03693 −79.21062 −79.17814 −79.20387 −79.20986 — — −79.20682

cc-pVDZ 1.310(+0.012) 107.0(−41.6) 1.303(+0.005) 173.2(+24.6) 1.289(−0.009) — 1.294(−0.004) 165.1(+16.5) 1.293(−0.005) 169.0(+20.4) 1.285(−0.013) — 1.298(±0.000) — 1.298 148.6

1432(−218) 1670(+19) 1702(+51) 1685(+34) 1694(+43) 1698(+47) 1640(−11) 1651

−79.07582 −79.28333 −79.25960 −79.28205 −79.28975 — — −79.26794

cc-pVTZ 1.304(+0.019) 114.3(−40.1) 1.283(−0.002) 181.3(+26.9) 1.273(−0.012) — 1.277(−0.008) 174.1(−19.8) 1.275(−0.010) 178.6(+24.2) 1.244(−0.008) — 1.284(−0.001) — 1.285 154.4

16.2(−7.7) 33.7(−0.2) — 45.5(+11.6) 46.7(+12.8) 69.2(+35.3) — 33.9

of 1.1 − 6.0 Å for the first excited state of the BN molecule. The points on the resulting potential energy curve were used for a subsequent generalized Morse function 64 fit to obtain the equilibrium bond lengths (Re ) and potential energy depths (De ). The harmonic vibrational frequencies (ωe ) were calculated numerically using the five-point finite difference stencil. 65

4 Numerical Results 4.1 Dissociation of C2 The carbon dimer is one of the most complex diatomic molecules that can be formed from the first-row elements of the periodic table. The unusual nature of the carbon-carbon bond and the question concerning its bond order attracted a lot of attention from theoretical chemists 69–72 in


ωe [cm−1 ]

1630(−52) 1751(+68) 1739(+57) 1734(+52) 1749(+67) 1732(+50) 1681(+1) 1682


0.00 −0.05 Energy [Eh]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

AP1roG AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD NEVPT2

−0.10 −0.15 −0.20 −0.25

1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 Distance [Å] Figure 1: Potential energy surfaces for the dissociation of the C2 molecule using an aug-cc-pVTZ basis set. See Fig. S1 of the Supporting Information for potential energy surfaces aligned at the equilibrium distance. recent years. Around the equilibrium distance, the electronic structure of the C2 molecule has two dominant configurations 1σg2 1σu2 2σg2 2σu2 1πu4 and 1σg2 1σu2 2σg2 1πu4 3σg2 as well as a number of other configurations arising from low-lying excited states. 66,73–75 When the two carbon atoms are pulled apart, the molecular system becomes strongly multi-reference. However, even for stretched carbon-carbon distances, dynamical electron correlation effects remain non-negligible. 46,73,75 A reliable theoretical description of spectroscopic constants (bond lengths, potential energy well depths, and vibrational frequencies) thus requires a balanced treatment of all types of electron correlation effects 45,76 (static, non-dynamic, and dynamic) along the dissociation pathway. Since highly accurate reference data for the spectroscopic constants of C2 is available, it is an ideal candidate to test our AP1roG-LCC approach. The upper part of Table 1 summarizes the spectroscopic constants of the C2 molecule for different basis sets and quantum chemistry methods including various dynamical correlation models


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based on an AP1roG reference function. As expected, AP1roG predicts too short equilibrium bond distances and too shallow potential well depths for all basis sets studied. If dynamical correlation is included a posteriori on top of the AP1roG reference function using perturbation theory, spectroscopic constants improve only slightly compared to MRCI-SD reference data. Although AP1roG-PTa predicts equilibrium bond distances that are in perfect agreement with MRCI-SD, the potential well depth is overestimated and the differences with respect to MRCI-SD are even larger than for AP1roG without PTa correction. Furthermore, AP1roG-PTb does not improve potential well depths and vibrational frequencies compared to AP1roG when the basis set is enlarged from aug-cc-pVDZ to aug-cc-pVTZ. In contrast to the PTa and PTb models, an LCC correction on top of AP1roG including doubles and singles and doubles yields spectroscopic constants that are in very good agreement with MRCI-SD reference data, outperforming NEVPT2 (differences are less than 2 kcal/mol for potential well depths using aug-cc-pVTZ). Figure 1 shows the fitted potential energy surfaces for the aug-cc-pVTZ basis set. We should note that all potential energy surfaces were shifted to zero in the dissociation limit. Additional plots of all potential energy surfaces aligned at the equilibrium distance are summarized in Figs. S1 to S3 of the Supporting Information. Comparing the spectroscopic constants of Table 1, we can conclude that the MRCI-SD potential energy surface would lie between the AP1roG-LCCD and AP1roG-LCCSD potential energy curves. All other quantum chemistry methods yield potential energy surfaces that deviate more from the expected MRCI-SD reference curve.

4.2 Dissociation of F2 F2 is a well-known example of diatomic molecules where dynamical electron correlation effects play a dominant role. 72 Furthermore, theoretical studies indicate that large basis sets and robust dynamical electron correlation models are required to reproduce the experimentally determined spectroscopic constants. 77–82 These features make the F2 molecule a good test case to assess the reliability of the LCC correction on top of an AP1roG reference function. The middle part of Table 1 summarizes the spectroscopic constants for the dissociation process 13 ACS Paragon Plus Environment


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of the F2 molecule using different basis sets and dynamical correlation models. We should note that both CCSD(T) and AP1roG-PTb diverge in the dissociation limit and thus the potential energy well depths are not given in Table 1 (see also Figure 2). To obtain an estimate for De , we have taken the MRCI-SD results by Peterson and co-workers. 66 Note that the re and ωe as predicted by MRCI-SD are in good agreement with CCSD(T) calculations. As observed for the C2 molecule, AP1roG considerably overestimates the equilibrium bond length and underestimates the potential energy depth, which can be attributed to the large fraction of dynamical correlation that cannot be captured by restricting the wavefunction to electron-pair states. Although AP1roG-PTa yields potential energy depths that are in very good agreement with MRCI-SD reference data, equilibrium bond lengths and vibrational frequencies deviate considerably from the MRCI-SD reference values. Specifically, when increasing the basis set from augcc-pVDZ to aug-cc-pVTZ, the predicted equilibrium bond length changes from being too short to being too long. In contrast to PTa, AP1roG-PTb results in equilibrium bond lengths and vibrational frequencies that are in perfect agreement with MRCI-SD reference data when the basis set is increased to triple-ζ quality, but it fails in the vicinity of dissociation. The LCC correction on top of AP1roG results in the most stable and robust dynamical correlation model, yielding similar results for increasing basis set sizes and outperforming CCSD. Specifically, equilibrium bond lengths and vibrational frequencies are in very good agreement with MRCI-SD reference data. However, potential energy well depths are considerably overestimated by AP1roG-LCC (around 12 kcal/mol with respect to MRCI-SD), which might be attributed to restricting the cluster operator to single and double excitations 77 (cf., CCSD overestimates De by more than 20-35 kcal/mol, depending on the basis set size). The differences in potential energy surfaces with respect to CCSD(T) reference data are summarized in Fig. S4 and Table S1 of the Supporting Information.

4.3 Dissociation of BN The BN molecule is isoelectronic to C2 and has been intensely studied both experimentally and theoretically. 67,68,71,83–89 It represents a computationally challenging molecule, primarily because 14 ACS Paragon Plus Environment

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0.00 −0.02 Energy [Eh]

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AP1roG AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD CCSD CCSD(T)

−0.04 −0.06 −0.08 −0.10 1.50

2.00

2.50

3.00

3.50

Distance [Å] Figure 2: Potential energy surfaces for the dissociation of the F2 molecule using a aug-cc-pVTZ basis set. See Fig. S2 of the Supporting Information for potential energy surfaces aligned at the equilibrium distance. the lowest lying electronic states possess a strong multi-reference character. The ground state of the BN molecule has been confirmed (both experimentally and theoretically) to be the 3 Π state, which can be assigned the 1σ 2 2σ 2 3σ 2 4σ 2 1π 3 5σ electronic configuration, and is very close in energy to the lowest-lying 1 Σ+ state, which can be characterized by a 1σ 2 2σ 2 3σ 2 4σ 2 1π 4 electronic configuration. Specifically, the 1 Σ+ excited state of BN dissociates to B(2 P)+N(2 D) which are higher in energy than the corresponding ground state atoms B(2 P)+N(4 S). Due to its strong multi-reference character, the first excited state of the BN molecule represents another good test case to assess the accuracy of AP1roG and the dynamical correlation models. The bottom part of Table 1 summarizes the spectroscopic constants for the dissociation process of the 1 Σ+ state of the BN molecule using different basis sets and dynamical correlation models. We should note that AP1roG-PTb diverges in the dissociation limit (for rBN ≥ 3.50 Å onwards, not shown in Figure 2) and thus De is not given in Table 1. To obtain an estimate for the potential



0.00 −0.05 Energy [Eh]

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−0.10 −0.15 −0.20

AP1roG AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD

−0.25 −0.30

1.50

2.00

2.50

3.00

3.50

Distance [Å] Figure 3: Potential energy surfaces for the dissociation of the first excited state of the BN molecule using a cc-pVTZ basis set. See Fig. S3 of the Supporting Information for potential energy surfaces aligned at the equilibrium distance. energy well depth, we have taken the MRCI-SD+Q results (including a multi-reference analog of the Davidson correction) by Peterson. 67 Note that re and ωe as predicted by MRCI-SD+Q are in good agreement with RMR CCSD(T) calculations by Li and Paldus. 68 As observed for the C2 and F2 molecules, AP1roG considerably overestimates the equilibrium bond length and underestimates the potential energy depth. Including dynamical correlation effects a posteriori using either perturbation theory or the LCC correction improves the errors in spectroscopic constants. The largest deviations from MRCI-SD+Q reference data are found for De which is overestimated by more than 16 kcal/mol. Furthermore, while AP1roG-PTa yields equilibrium bond lengths that are very good agreement with MRCI-SD+Q, AP1roG-LCCD and AP1roG-LCCSD result in smaller errors for the potential well depth. Specifically, when increasing the basis set from cc-pVDZ to cc-pVTZ, re predicted by AP1roG-PTa changes from being too short to being overestimated. Similar trends are found for the F2 molecule. In contrast to C2


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and F2 , AP1roG-PTb results in spectroscopic constants that deviate the most from MRCI-SD+Q. As observed for homonuclear diatomics, the LCC correction based on an AP1roG reference function results in the most stable and robust dynamical correlation model, yielding similar results for increasing basis set sizes and outperforming CCSD(T). The potential energy surfaces for the dissociation pathway of the 1 Σ state of BN are shown in Figure 3. Specifically, AP1roG predicts a rather flat and elongated dissociation curve. The shape of the potential energy curve changes considerably when adding dynamical correlation effects a posteriori, predicting a steeper dissociation pathway. We should note that all the dynamical correlation models investigated in this work yield similar potential energy surfaces (c.f., Table 1).

4.4 Symmetric Dissociation of H50 The symmetric stretching of hydrogen chains is commonly used as a molecular model for strongly correlated systems and remains a challenging problem for conventional quantum-chemistry methods. 37,42,90–93 Recently, we have shown that AP1roG accurately describes the potential energy surface of the symmetric dissociation of H50 in the vicinity of dissociation, but deviates from DMRG reference data around equilibrium and for stretched interatomic distances, 37 which can be attributed to the missing dynamical correlation energy. Table 2 summarizes the non-parallelity error (NPE) per hydrogen atom for the symmetric dissociation of H50 obtained by AP1roG and different dynamical correlation models. The large NPE per hydrogen atom of AP1roG can be associated with the missing dynamical correlation energy around the equilibrium distance and for stretched interatomic bond lengths. Adding dynamical correlation a posteriori improves the NPE per hydrogen atom considerably. While PTa, PTb, and LCCD have a similar NPE per hydrogen atom of about 4.5 mEh , the NPE is reduced to less than 1.5 mEh if single excitations are included in the cluster operator. The importance of single excitations in the LCC model is also noticeable in the shape of the potential energy surface shown in Figure 4. While AP1roG-PTa, AP1roG-PTb, and AP1roGLCCD have similar potential energy curves (in terms of shape and total electronic energies) with 17 ACS Paragon Plus Environment


AP1roG-PTb being lowest in energy, the potential energy surface obtained by AP1roG-LCCSD considerably deviates from the aforementioned dynamical correlation models for short and intermediate bond lengths (around 1.0 and 1.5 Å). Furthermore, the AP1roG-LCCSD potential energy curve is in very good agreement with DMRG reference data (see Figure 4).

−23.50 −24.00 −24.50 Energy [Eh]

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−25.00 −25.50 −26.00

DMRG MP2 AP1roG AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD

−26.50 −27.00 0.60

0.80

1.00

1.20 1.40 Distance [Å]

1.60

1.80

2.00

Figure 4: Symmetric dissociation of the H50 chain using the STO-6G basis set obtained from different methods. The DMRG reference data are taken from Ref., 90 while the MP2 and AP1roG data are taken from Ref. 37 .

4.5 Symmetric Dissociation of UO2+ 2 The uranyl cation (UO2+ 2 ) is a small building block of a large variety of uranium-containing complexes. 94–96 This molecule has a linear structure and a singlet ground-state electronic configuration. Its characteristic symmetric and asymmetric U–O vibrational frequencies are used to identify 95,97,98 While the electronic structure of the the presence of UO2+ 2 in larger molecular assemblies.

uranyl cation is well-understood around the equilibrium structure, 47,99–108 the complicated nature 18 ACS Paragon Plus Environment

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Table 2: Non-parallelity error per hydrogen atom for the symmetric dissociation of the H50 chain with respect to DMRG reference data. The MP2 and AP1roG data are taken from Ref. 37 Method MP2 AP1roG AP1roG-PTa AP1roG-PTb AP1roG-LCCD AP1roG-LCCSD

NPE/H [mEh ] 43.540 6.187 4.831 4.170 4.506 1.389

Table 3: Spectroscopic constants for the symmetric dissociation of the UO2+ 2 molecule for different quantum chemistry. The differences are with respect to CCSD(T) reference data. The CASSCF and CC data were taken from Ref. 47. Method AP1roG AP1roG-PTb AP1roG-LCCD AP1roG-LCCSD CAS(10,10)SCF CAS(12,12)SCF CCD CCSD CCSD(T)

re [Å] 1.669(−0.047) 1.715(−0.001) 1.712(−0.004) 1.724(+0.008) 1.694(−0.022) 1.707(−0.009) 1.690(−0.026) 1.697(−0.019) 1.716

ωe [cm−1 ] 1062(+53) 1340(+331) 997(−12) 1027(+18) 1079(+70) 1034(+25) 1125(+116) 1068(+59) 1009

of the U–O bond hampers a theoretical description at larger U–O distances using standard quantum chemistry approaches. 47 One of the limiting factors that impede theoretical studies is the large number of strongly-correlated electrons distributed among the 5 f -, 6d-, and 7s-orbitals. In addition, the 6s- and 6p-core-valence orbitals are easily polarizable and have a non-negligible contribution to the correlation energy. However, around the equilibrium structure, the uranyl cation is well described by single-reference CC theory if all important electrons are correlated. This allows us to assess the performance of the LCC models in describing dynamical correlation effects originating from the 5 f -, 6d-, and 7s- as well as the core-valence electrons. The equilibrium bond lengths and vibrational frequencies obtained by different quantum chemistry methods are shown in Table 3. As expected, AP1roG considerably underestimates the equi19 ACS Paragon Plus Environment


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librium bond length, while ωe is in good agreement with CCSD(T). Adding dynamical correlation effects on top of AP1roG shifts re closer to the CCSD(T) reference data. The shape of the potential, however, strongly depends on the AP1roG dynamical correlation model. Specifically, PTb results in a much steeper potential energy surface overestimating vibrational frequencies by more than 330 cm−1 compared to CCSD(T), while LCCD and LCCSD preserve the shape of the potential energy surface and yield a vibrational frequency that agree well with AP1roG and CCSD(T) data (differences amount to approximately 20 cm−1 ). The overall accuracy of AP1roG-LCC lies between CCSD and CCSD(T), being closer to the latter. We should emphasize that PTa completely fails for the UO2+ 2 molecule and produces a discontinuous potential energy surface around equilibrium (see also Figure 5). Furthermore, the CASSCF equilibrium distance strongly depends on the size of the active space chosen in CASSCF calculations. Specifically, increasing the active space from CAS(10,10) to CAS(12,12), i.e., including the σ - and σ ∗ -orbitals, results in spectroscopic constants that are in good agreement with AP1roG-LCCD and CCSD(T) data. Figure 5 shows the fitted potential energy surfaces around equilibrium for selected quantum chemistry methods. AP1roG-LCC yields total electronic energies that are between CCSD and CCSD(T), while the potential energy surface predicted by AP1roG-PTb is considerably lower than the CCSD(T) reference curve. Note that the potential energy surfaces optimized by CASSCF lie much higher in energy and are thus not shown in Figure 5. The differences in potential energy surfaces with respect to CCSD(T) reference data are summarized in Fig. S6 and Table S3 of the Supporting Information.

5 Conclusions We have presented an alternative model to capture dynamical correlation effects on top of an AP1roG reference functions that uses a Linearized Coupled Cluster ansatz. Our approach is motivated by previous studies of an LCC correction to an APSG reference function. 55 Specifically, our cluster operator is restricted to double and single and double excitations as in the standard Coupled


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CCD CCSD CCSD(T) AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD

−625.44 −625.46 −625.48 Energy [Eh]

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−625.50 −625.52 −625.54 −625.56 −625.58 −625.60

1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 Distance [Å]

Figure 5: Potential energy surfaces for the symmetric stretching of the UO2+ 2 molecule around the equilibrium geometry. Note that the CASSCF potential energy surfaces are much higher in energy and are thus not shown. Cluster approach, i.e., excitations from the occupied to the virtual orbitals of some reference determinant. We have compared this new dynamical correlation ansatz to the PTa and PTb perturbation theory models as well as standard quantum chemistry approaches for the C2 and F2 molecules, the first excited state of the BN molecule, the H50 hydrogen chain, and the uranyl cation, UO2+ 2 . We should note that our LCC corrections are similar to fpCC methods, where first the pair amplitudes are optimized and then the equations for the remaining non-pair amplitudes are solved, while the pair amplitudes are kept fixed. In contrast to fpCC, the cluster operator is linearized and |AP1roGi is chosen as the reference function. This leads to additional terms in the amplitude equations in contrast to the standard single-reference linearized CC approach, but fewer terms than in the fpCC method because of the missing higher-order terms of the non-pair amplitudes. The scaling of both LCC corrections is similar to CCD/CCSD, O(o2 v4 ) (o: occupied, v: virtual). Thus, AP1roG-LCC is computationally more expensive than AP1roG-PT, which scales as O(K 5 ), where 21 ACS Paragon Plus Environment


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K is the number of basis functions. In general, both AP1roG-LCC models yield similar spectroscopic constants and are closest to MRCI-SD, CCSD(T), and DMRG reference data for all molecules we have investigated. Furthermore, LCCD and LCCSD represent more robust and reliable dynamical correlation models than PTa and PTb. Our study demonstrates that the success and failures of PTa and PTb are difficult to anticipate a priori and are strongly system-dependent. While PTa yields reasonable results for equilibrium bond lengths of the C2 molecule and for the potential energy depth of the F2 molecule, the corresponding C2 potential energy depth and the F2 equilibrium bond length significantly differ from reference data. A similar behavior was observed for PTb. In contrast to the perturbation theory models, the LCC ansatz is able to capture different flavours of dynamical correlation effects reliably, as present in the C2 , F2 , BN, H50 , and UO2+ 2 molecules. Furthermore, the LCC models presented in this work account for dynamical correlation effects a posteriori, i.e., the missing dynamical correlation effects that cannot be described by electron pairs. The orbital basis employed in the LCC corrections is, however, optimized using the AP1roG method and hence already includes a (small, but unknown) fraction of dynamical correlation effects. Adding dynamical correlation on top of AP1roG may thus lead to double-counting of dynamical correlation effects. Moreover, our LCC models are equivalent to a MR-Coupled Electron Pair Approximation(0) (CEPA(0)) treatment, which is known to overestimate the electron correlation energy. Our laboratory is currently investigating the effect of orbital rotations, the extent of double-counting, as well as the tendency to overestimate electron correlation. Finally, we should emphasize that LCC is usually susceptible to divergences due to singularities in the amplitude equations, which can be redressed using regularization techniques. 109 For the molecules studied here, we did not encounter any problems with divergences when optimizing the LCC amplitudes. However, this might not be true for other systems. For example, the N2 molecule is a particularly challenging system for electron-pair theories and a posteriori dynamical correlation models. 49,58 The performance of the LCC corrections and possible divergence problems in the LCC amplitude equations are currently being investigated in our laboratory, and will be the subject


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of a future publication. Nonetheless, our work suggest that AP1roG-LCC represents a computationally feasible and promising approach to account for dynamical correlation effects on top of AP1roG a posteriori. Further computational studies using a wider set of molecules are, however, required to scrutinize the performance and numerical stability of AP1roG-LCC.

6 Acknowledgment We gratefully acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada. K.B. acknowledges the financial support from the Swiss National Science Foundation (P2EZP2 148650), the Banting Postdoctoral Fellowship program, and the National Science Center (Grant No. DEC-2013/11/B/ST4/00771). We had many helpful discussions with Paweł Tecmer and Ireneusz Grabowski.

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Graphical TOC Entry


0.00

Energy [Eh]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Journal of Chemical Theory and ComputationPage 32 of 37

−0.05 −0.10 −0.15

AP1roG AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD NEVPT2

−0.20 −0.25 ACS Paragon Plus Environment

1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 Distance [Å]

Page 33 of 37Journal of Chemical Theory and Computation

0.00

Energy [Eh]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

−0.02 AP1roG AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD CCSD CCSD(T)

−0.04 −0.06 −0.08 −0.10

ACS Paragon Plus Environment

1.50


3.00

3.50

0.00

Energy [Eh]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Journal of Chemical Theory and ComputationPage 34 of 37

−0.05 −0.10 −0.15 −0.20

AP1roG AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD

−0.25 −0.30


1.50


3.00

3.50

Page−23.50 35 of 37

Energy [Eh]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21


−24.00 −24.50 −25.00 −25.50 −26.00

DMRG MP2 AP1roG AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD

−26.50 −27.00 0.60

0.80


1.00


1.60

1.80

2.00


−625.44

Energy [Eh]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

−625.46 −625.48

CCD CCSD CCSD(T) AP1roG−PTa AP1roG−PTb AP1roG−LCCD AP1roG−LCCSD

Page 36 of 37

−625.50 −625.52 −625.54 −625.56 −625.58 −625.60


1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 Distance [Å]

Page Journal 37 ofof 37Chemical Theory and Computation

1 2 3 4


Linearized Coupled Cluster Correction on the Antisymmetric Product

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